30. Passage of particles through matter 1 - Particle Data Group

30. Passage of particles through matter 17
x/2
x
s plane
Ψ plane
y plane
θ plane
Figure 30.10: Quantities used to describe multiple Coulomb scattering.
The particle is incident in the plane of the figure.
The nonprojected (space) and projected (plane) angular distributions are given
approximately by [35]
1
⎧
exp ⎪⎩− θ2 ⎫
space
⎪⎭dΩ , (30.16)
2π θ 2 0
1
√ 2π θ0
exp
⎧
2θ 2 0
⎪⎩− θ2 plane
2θ 2 0
⎫
⎪⎭dθplane , (30.17)
where θ is the deflection angle. In this approximation, θ 2 space ≈ (θ 2 plane,x + θ2 plane,y ),
where the x and y axes are orthogonal to the direction of motion, and
dΩ ≈ dθplane,x dθplane,y. Deflections into θplane,x and θplane,y are independent and
identically distributed.
Fig. 30.10 shows these and other quantities sometimes used to describe multiple
Coulomb scattering. They are
ψ rms 1
plane = √ θ
3 rms
plane
y rms 1
plane = √ xθ
3 rms
plane
= 1
√ 3 θ0 , (30.18)
= 1
√ 3 xθ0 , (30.19)
s rms 1
plane =
4 √ rms 1
xθ plane =
3 4 √ 3 xθ0 . (30.20)
All the quantitative estimates in this section apply only in the limit of small
θ rms
plane and in the absence of large-angle scatters. The random variables s, ψ, y, and
θ in a given plane are correlated. Obviously, y ≈ xψ. In addition, y and θ have the
correlation coefficient ρyθ = √ 3/2 ≈ 0.87. For Monte Carlo generation of a joint
(y plane,θplane) distribution, or for other calculations, it may be most convenient
to work with independent Gaussian random variables (z1,z2) with mean zero and
variance one, and then set
yplane =z1 xθ0(1 − ρ 2 yθ )1/2 / √ 3 + z2 ρyθxθ0/ √ 3 (30.21)
June 18, 2012 16:19